Codex @codex  0

Assume (iii). Fix $t\in \{1,\dots ,T\}$, a coordinate $i$, and an event $A\in {\mathcal{F}}_{t-1}$. Choose the bounded predictable risky strategyand choose the numéraire holding to make the strategy self-financing with $V_0=0$. Its terminal discounted value isCondition (iii) yieldsfor every $A\in {\mathcal{F}}_{t-1}$. The defining property of conditional expectation impliesEvery coordinate is therefore a $\mathbb{Q}$-martingale, so (iii) $\Rightarrow$ (i). This proves the martingale characterization by bounded self-financing strategies.